Circular Cylinder Problems, 3

Category: Solid Geometry

"Published in Newark, California, USA"

A circular concrete conduit, whose inside diameter is 10 ft., is 1 ft. thick. It rises 16 ft. per 1000 horizontal feet. The vertical plane which contains the axis is perpendicular to the two vertical planes which contain the ends of the conduit. If the ends are 3000 ft. apart, find the amount of concrete used in the construction of the conduit.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The first thing that we need to do is to solve for the altitude or vertical distance of the ends of a conduit which is a right circular cylinder by similar triangles as follows

By Pythagorean Theorem, the length of a concrete conduit is

 

 

 

 

The cross section of a concrete conduit is a two concentric circles. The area of the region between the two concentric circles is
 

 

 

Therefore, the amount of a concrete used for the construction of a conduit which is the volume of a circular cylinder is

Circular Cylinder Problems, 2

Category: Solid Geometry

"Published in Newark, California, USA"

A straight stairway has a balustrade, with dimensions as shown, containing 2 circular balustrade to a step. The foot and hand rails are parallel. Find the total amount of wood used and the total area to be painted.

Photo by Math Principles in Everyday Life

Solution:

Consider the given figure above

Photo by Math Principles in Everyday Life

A balustrade in a stairway is in a shape of right circular  cylinder. Each step has two balustrades and there are 14 steps in a stairway. Hence, there are 28 balustrades in a stairway. 

The surface area of a right circular cylinder which is a balustrade is

Therefore, the total area of balustrades to be painted is

The volume of a right circular cylinder which is a balustrade is

Therefore, the amount of wood used for balustrades is

 

Circular Cylinder Problems

Category: Solid Geometry

"Published in Newark, California, USA"

A pine log 40 ft. long has a uniform circular cross section of radius 11 in. The ends lie in parallel planes which are not perpendicular to the axis of the log. How much paint will be required to give the log (ends not included) two coats, if 1 gal. of paint covers 800 sq. ft. of surface? If pine weighs 31 lbs. per cu. ft., find the weight of the log.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The surface or lateral area of a circular cylinder which is a pine log is

where p_k is the circumference or perimeter of the cross section and e is the length of an element or the slant height. Substitute the values, we have

Therefore, the amount of paint required to paint the surface of a pine logs in two coats is

The volume of a circular cylinder which is a pine log is

where K is the area of a cross section and e is the length of an element or the slant height. Substitute the values, we have
 

 

 

 

 

Therefore, the weight of a pine log is